No abstract

Clustering behaviours have been found in numerous multi-strain transmission models. Numerical solutions of these models have shown that steady-states, periodic, or even chaotic motions can be self-organized into clusters. Such clustering behaviours are not a priori expected. It has been proposed that the cross-protection from multiple strains of pathogens is responsible for the clustering phenomenon. In this paper, we show that the steady-state clusterings in existing models can be analytically predicted. The clusterings occur via semi-simple double zero bifurcation from the quotient networks of the models and the patterns which follow can be predicted through the stability analysis of the bifurcation. We calculate the stability criteria for the clustering patterns and show that some patterns are inherently unstable. Finally, the biological implications of these results are discussed.

Investigating a method of chaos control for one-dimensional maps, where the intervention is proportional to the difference between a fixed value and a current state, we demonstrate that stabilization is possible in one of the two following cases: (1) for small values, the map is increasing and the slope of the line connecting the points on the line with the origin is decreasing; (2) the chaotic map is locally Lipschitz. Moreover, in the latter case we prove that any point of the map can be stabilized. In addition, we study pulse stabilization when the intervention occurs each m-th step and illustrate that stabilization is possible for the first type of maps. In the context of population dynamics, we notice that control with a positive target, even if stabilization is not achieved, leads to persistent solutions and prevents extinction in models which experience the Allee effect.

In this paper, we will analyze the bifurcation dynamics of an in vivo model of Plasmodium falciparum. The main attention of this model is focused on the dynamics of cross-reactivity from antigenic variation. We apply the techniques of coupled cell systems to study this model. It is shown that synchrony-breaking Hopf bifurcation occurs from a nontrivial synchronous equilibrium. In proving the existence of a Hopf bifurcation, we also discover the condition under which possible 2-color synchrony patterns arise from the bifurcation. The dynamics resulting from the bifurcation are qualitatively similar to known behavior of antigenic variation. These results are discussed and illustrated with specific examples and numerical simulations.

Published accounts of pedagogical experience and pedagogical research are critical resources to post-secondary mathematics instructors, and yet the quantity and scope of this literature is rarely summarized or reviewed. In this contribution, we analyze recent peer-reviewed journal publications regarding post-secondary mathematics, published by Canadian scholars. We classified this scholarship by institution, publication year, type of pedagogical scholarship, and by topic. We highlight topics of continual interest, changing trends in time and newly emerging themes. This review therefore provides a benchmark of current scholarship in this important area, as well as a point of comparison for similar data from other countries, and other disciplines. Les comptes rendus publiés sur les expériences pédagogiques et la recherche sur la pédagogie sont des ressources essentielles pour les enseignants de mathématiques au niveau postsecondaire. Pourtant, la quantité et la portée de cette documentation font rarement l’objet de résumés ou d’analyses. Dans cet article, nous analysons les publications récentes de chercheurs canadiens sur les mathématiques au niveau postsecondaire, qui ont paru dans des revues révisées par les pairs. Nous avons classé ces publications par établissement, année de publication, type de recherche et sujet. Nous mettons en lumière les sujets d’intérêt constant, les tendances en évolution au fil du temps et les thèmes émergents. Cette recension constitue donc une référence sur les recherches universitaires actuelles dans ce domaine important ainsi qu’un point de comparaison pour des données similaires provenant d’autres pays et d’autres disciplines.